the rate of progress of evolution - simons institute for the ...nicholas pippenger the rate of...

The Rate of Progress of Evolution

Nicholas Pippenger

20 March 2014

Nicholas Pippenger The Rate of Progress of Evolution

Progress—Goals

I There will be goals.

I But they will not be imputed to nature.

Progress—Goals

My Goals (for this talk)

I Ultimately, quantitative estimates of “rates”.

I But “progress” might not be proportional to time.

I Presently, functional forms.

I But don’t lump all power laws together as “polynomial”.

I Emphasize negative results.

“The Cost of Natural Selection”

I J. B. S. Haldane, J. Genetics,1957.

I About 10—300 generations per substitution.

I J. Maynard Smith, Nature, 1968.

I Truncation.

Assumptions

I Fix n alleles (initially present in moderate amounts).

I Haploid genome.

I No mutation.

I Non-overlapping generations.

I Sexual reproduction.

I No assortative mating.

I No linkage in recombination.

I Fecundity unaffected.

I Omniscience as to genome.

I Omnipotence as to opportunity for reproduction.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Assumptions

I Haploid genome.

I No mutation.

Basic Step

I Choose a threshold.

I Truncate population to individuals above threshold.

I Let population regain its original size.

Basic Step

Conclusions

I Iteration of the basic step is essentially optimal.

I #generations ≥ C√

I This is (a lower bound for) the true “cost of naturalselection”.

Conclusions

What If There’s No Sex?

I #generations ≥ C n.

What If There’s No Sex?

I #generations ≥ C n.

Evolution As a Random Walk

I S. J. Gould, Full House, 1996.

Evolution As a Random Walk

I S. J. Gould, Full House, 1996.

A Random Walk on a Line

• •p

oo1−p

// •

I p = 1/2.

I At time T , most walkers are to the left of C√

• •p

oo1−p

// •

I p = 1/2.

• •p

oo1−p

// •

I p = 1/2.

• •p

oo1−p

// •

I p = 1/2.

Objection

I Sensitive to parameter: p = 1/2.

I If p < 1/2, populations sweeps to right.

I If p > 1/2, distribution approaches exponential.

Objection

A Random Walk in a Random Environment

I Create a random environment.

I Take a random walk in that environment.

The Random Environment

I An infinite tree, created by a branching process.

I Pr[two children] = a, Pr[no children] = 1− a.

I Condition on non-extinction.

◦ ◦

• •

ppppppppppppp

NNNNNNNNNNNNN

◦ ◦

1− a a

The Random Walk

I Pr[ascend] = p.

I At a leaf, Pr[stay] = 1− p.

I At a binary node, Pr[descend left] = Pr[descend right] = 1−p2 .

The Random Walk

I Pr[ascend] = p.

The Random Walk

I Pr[ascend] = p.

The Random Walk

I Pr[ascend] = p.

The Random Walk

◦ ◦

ZZ •

1−p2

xxqqqqqqq1−p2

&&MMMMMMM

◦ ◦

I Gould: a = 1.

I H. Kesten, Ann. Inst. H. Poincare, 1986.

I a = 1/2, p = 1/3.

I At time T , most walkers are above level C T 1/3.

I Gould: a = 1.

I a = 1/2, p = 1/3.

I Gould: a = 1.

I a = 1/2, p = 1/3.

I Gould: a = 1.

I a = 1/2, p = 1/3.

I Gould: a = 1.

I a = 1/2, p = 1/3.

Trapping

I Traps are entered with fixed probability 1/2 at each level.

I Time to escape a random trap is finite with probability one.

I But its expectation is infinite.

Trapping

Trapping occurs if 1/2 < a < 1 and 0 < p < (1− a)/(2− a).

1/2 • •

1/3 •

iiiiiiiiiiiiiiiiiiiiiiiiiii

0 • •

1/2 a 1

Trapping occurs if 1/2 < a < 1 and 0 < p < (1− a)/(2− a).

1/2 • •

1/3 •

iiiiiiiiiiiiiiiiiiiiiiiiiii

0 • •

1/2 a 1

At time T , most walkers are above level C T θ, where

θ ≤log 1

log 14p

I Trapping is different from diffusion against a wall.

I Trapping is exacerbated by positive bias.

θ ≤log 1

log 14p

θ ≤log 1

log 14p

Thank You

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